If $H$ is a Hilbert space and $T$ and $S$ are positive operators on $H$ (i.e. $ \ge 0$ and $ \ge 0$, for every $y \in H$ ) such that $T \le S$ (or $0 \le S-T$). Does it exist necessarily a positive constant $r$ such that $T^2 \le r^2 S^2$ ?. Notice that this need not hold for $r = 1$.